A note on sparse least-squares regression

We compute a \emph{sparse} solution to the classical least-squares problem $\min_x||A x -b||,$ where $A$ is an arbitrary matrix. We describe a novel algorithm for this sparse least-squares problem. The algorithm operates as follows: first, it selects columns from $A$, and then solves a least-squares problem only with the selected columns. The column selection algorithm that we use is known to perform well for the well studied column subset selection problem. The contribution of this article is to show that it gives favorable results for sparse least-squares as well. Specifically, we prove that the solution vector obtained by our algorithm is close to the solution vector obtained via what is known as the "SVD-truncated regularization approach".Comment: Information Processing Letters, to appea

Automatic alignment for three-dimensional tomographic reconstruction

In tomographic reconstruction, the goal is to reconstruct an unknown object from a collection of line integrals. Given a complete sampling of such line integrals for various angles and directions, explicit inverse formulas exist to reconstruct the object. Given noisy and incomplete measurements, the inverse problem is typically solved through a regularized least-squares approach. A challenge for both approaches is that in practice the exact directions and offsets of the x-rays are only known approximately due to, e.g. calibration errors. Such errors lead to artifacts in the reconstructed image. In the case of sufficient sampling and geometrically simple misalignment, the measurements can be corrected by exploiting so-called consistency conditions. In other cases, such conditions may not apply and we have to solve an additional inverse problem to retrieve the angles and shifts. In this paper we propose a general algorithmic framework for retrieving these parameters in conjunction with an algebraic reconstruction technique. The proposed approach is illustrated by numerical examples for both simulated data and an electron tomography dataset

A fast Total Variation-based iterative algorithm for digital breast tomosynthesis image reconstruction:

In this work, we propose a fast iterative algorithm for the reconstruction of digital breast tomosynthesis images. The algorithm solves a regularization problem, expressed as the minimization of the sum of a least-squares term and a weighted smoothed version of the Total Variation regularization function. We use a Fixed Point method for the solution of the minimization problem, requiring the solution of a linear system at each iteration, whose coefficient matrix is a positive definite approximation of the Hessian of the objective function. We propose an efficient implementation of the algorithm, where the linear system is solved by a truncated Conjugate Gradient method. We compare the Fixed Point implementation with a fast first order method such as the Scaled Gradient Projection method, that does not require any linear system solution. Numerical experiments on a breast phantom widely used in tomographic simulations show that both the methods recover microcalcifications very fast while the Fixed Point is more efficient in detecting masses, when more time is available for the algorithm execution